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November  2016, 15(6): 2059-2074. doi: 10.3934/cpaa.2016027

On the Hardy-Littlewood-Sobolev type systems

Ze Cheng 1, , Genggeng Huang 2, and  Congming Li 3,

1. 

Department of Applied Mathematics, University of Colorado at Boulder, Colorado
2. 

Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China
3. 

Department of Applied Mathematics, University of Colorado at Boulder

Received  October 2015 Revised  June 2016 Published  September 2016

In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.
Keywords: existence, uniqueness., non-existence, Hardy-Littlewood-Sobolev.
Mathematics Subject Classification: Primary: 35J9.
Citation: Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027
References:
[1]

Disc. & Cont. Dynamics Sys., 34 (2014), 3317-3339. doi: 10.3934/dcds.2014.34.3317.  Google Scholar

[2]

Indiana University Mathematics Journal, 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012.  Google Scholar

[3]

Comm. Pure Appl. Math., 142 (1989), 615-622. doi: 10.1002/cpa.3160420304.  Google Scholar

[4]

Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[5]

Commun. in Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[6]

Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[7]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().   Google Scholar

[8]

Nonlinear Analysis: Theory, Methods and Applications, 114 (2015), 2-12. doi: 10.1016/j.na.2014.10.019.  Google Scholar

[9]

Math. Anal. and Applications, Part A, Advances in Math. Suppl. Studies, 7A (1981), 369-402.  Google Scholar

[10]

Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315. doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[11]

Calc. Var. of Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7.  Google Scholar

[12]

Comm. in Partial Differential Equation, 41 (2016), 1029-1039. doi: 10.1080/03605302.2016.1190376.  Google Scholar

[13]

Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[14]

Journal of Partial Differential Equations, 19 (2006), 256.  Google Scholar

[15]

Quaderno Matematico, (1982), 285.  Google Scholar

[16]

Communications in partial differential equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar

[17]

Differ. Integral Equations, 9 (1996), 465-479.  Google Scholar

[18]

Soviet Math. Doklady, 6 (1965), 1408-1411.  Google Scholar

[19]

Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[20]

Indiana Univ. J. Math., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[21]

Springer, 2007.  Google Scholar

[22]

Arch. Rat. Mech. Anal., 43 (1971), 304-318.  Google Scholar

[23]

Differ. Integral Equations, 9 (1996), 635-654.  Google Scholar

[24]

Atti Semi. Mat. Fis. Univ. Modena, 46 (1998), 369-380.  Google Scholar

[25]

Advances in Mathematics, 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[26]

Math. Ann., (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar

show all references

References:
[1]

Disc. & Cont. Dynamics Sys., 34 (2014), 3317-3339. doi: 10.3934/dcds.2014.34.3317.  Google Scholar

[2]

Indiana University Mathematics Journal, 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012.  Google Scholar

[3]

Comm. Pure Appl. Math., 142 (1989), 615-622. doi: 10.1002/cpa.3160420304.  Google Scholar

[4]

Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[5]

Commun. in Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[6]

Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[7]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().   Google Scholar

[8]

Nonlinear Analysis: Theory, Methods and Applications, 114 (2015), 2-12. doi: 10.1016/j.na.2014.10.019.  Google Scholar

[9]

Math. Anal. and Applications, Part A, Advances in Math. Suppl. Studies, 7A (1981), 369-402.  Google Scholar

[10]

Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315. doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[11]

Calc. Var. of Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7.  Google Scholar

[12]

Comm. in Partial Differential Equation, 41 (2016), 1029-1039. doi: 10.1080/03605302.2016.1190376.  Google Scholar

[13]

Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[14]

Journal of Partial Differential Equations, 19 (2006), 256.  Google Scholar

[15]

Quaderno Matematico, (1982), 285.  Google Scholar

[16]

Communications in partial differential equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar

[17]

Differ. Integral Equations, 9 (1996), 465-479.  Google Scholar

[18]

Soviet Math. Doklady, 6 (1965), 1408-1411.  Google Scholar

[19]

Duke Math. J., 139 (2007), 555-579. doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[20]

Indiana Univ. J. Math., 35 (1986), 681-703. doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[21]

Springer, 2007.  Google Scholar

[22]

Arch. Rat. Mech. Anal., 43 (1971), 304-318.  Google Scholar

[23]

Differ. Integral Equations, 9 (1996), 635-654.  Google Scholar

[24]

Atti Semi. Mat. Fis. Univ. Modena, 46 (1998), 369-380.  Google Scholar

[25]

Advances in Mathematics, 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[26]

Math. Ann., (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar

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